Abstract
For a computable field F, the splitting set S F of F is the set of polynomials with coefficients in F which factor over F, and the root set R F of F is the set of polynomials with coefficients in F which have a root in F.
Results of Frohlich and Shepherdson in [3] imply that for a computable field F, the splitting set S F and the root set R F are Turing-equivalent. Much more recently, in [5], Miller showed that for algebraic fields, the root set actually has slightly higher complexity: for algebraic fields F, it is always the case that S F ≤ 1 R F , but there are algebraic fields F where we have \(R_F \nleq_1 S_F\).
Here we compare the splitting set and the root set of a computable algebraic field under a different reduction: the weak truth-table reduction. We construct a computable algebraic field for which \(R_F \nleq_{wtt} S_F\).
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Steiner, R.M. (2010). Computable Fields and Weak Truth-Table Reducibility. In: Ferreira, F., Löwe, B., Mayordomo, E., Mendes Gomes, L. (eds) Programs, Proofs, Processes. CiE 2010. Lecture Notes in Computer Science, vol 6158. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13962-8_44
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DOI: https://doi.org/10.1007/978-3-642-13962-8_44
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